Ionic liquid gated field-effect transistors (FETs) based on semiconducting transition metal dichalcogenides (TMDs) are used to study a rich variety of extremely interesting physical phenomena, but important aspects of how charge carriers are accumulated in these systems are not understood. We address these issues by means of a systematic experimental study of transport in monolayer MoSe2 and WSe2 as a function of magnetic field and gate voltage, exploring accumulated densities of carriers ranging from approximately 10(14) cm(-2) holes in the valence band to 4 x 10(14) cm(-2) electrons in the conduction band. We identify the conditions when the chemical potential enters different valleys in the monolayer band structure (the K and Q valley in the conduction band and the two spin-split K-valleys in the valence band) and find that an independent electron picture describes the occupation of states well. Unexpectedly, however, the experiments show very large changes in the device capacitance when multiple valleys are occupied that are not at all compatible with the commonly expected quantum capacitance contribution of these systems, C-Q = e(2)/ (d mu/dn). A theoretical analysis of all terms responsible for the total capacitance shows that under general conditions a term is present besides the usual quantum capacitance, which becomes important for very small distances between the capacitor plates. This term, which we call cross quantum capacitance, originates from screening of the electric field generated by charges on one plate from charges sitting on the other plate. The effect is negligible in normal capacitors but large in ionic liquid FETs because of the atomic proximity between the ions in the gate and the accumulated charges on the TMD, and it accounts for all our experimental observations. Our findings therefore reveal an important contribution to the capacitance of physical systems that had been virtually entirely neglected until now.